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Theorem prdstset 17353
Description: Structure product topology. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
Hypotheses
Ref Expression
prdsbas.p 𝑃 = (𝑆Xs𝑅)
prdsbas.s (πœ‘ β†’ 𝑆 ∈ 𝑉)
prdsbas.r (πœ‘ β†’ 𝑅 ∈ π‘Š)
prdsbas.b 𝐡 = (Baseβ€˜π‘ƒ)
prdsbas.i (πœ‘ β†’ dom 𝑅 = 𝐼)
prdstset.l 𝑂 = (TopSetβ€˜π‘ƒ)
Assertion
Ref Expression
prdstset (πœ‘ β†’ 𝑂 = (∏tβ€˜(TopOpen ∘ 𝑅)))

Proof of Theorem prdstset
Dummy variables π‘Ž 𝑐 𝑑 𝑒 𝑓 𝑔 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prdsbas.p . . 3 𝑃 = (𝑆Xs𝑅)
2 eqid 2733 . . 3 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
3 prdsbas.i . . 3 (πœ‘ β†’ dom 𝑅 = 𝐼)
4 prdsbas.s . . . 4 (πœ‘ β†’ 𝑆 ∈ 𝑉)
5 prdsbas.r . . . 4 (πœ‘ β†’ 𝑅 ∈ π‘Š)
6 prdsbas.b . . . 4 𝐡 = (Baseβ€˜π‘ƒ)
71, 4, 5, 6, 3prdsbas 17344 . . 3 (πœ‘ β†’ 𝐡 = Xπ‘₯ ∈ 𝐼 (Baseβ€˜(π‘…β€˜π‘₯)))
8 eqid 2733 . . . 4 (+gβ€˜π‘ƒ) = (+gβ€˜π‘ƒ)
91, 4, 5, 6, 3, 8prdsplusg 17345 . . 3 (πœ‘ β†’ (+gβ€˜π‘ƒ) = (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(+gβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))))
10 eqid 2733 . . . 4 (.rβ€˜π‘ƒ) = (.rβ€˜π‘ƒ)
111, 4, 5, 6, 3, 10prdsmulr 17346 . . 3 (πœ‘ β†’ (.rβ€˜π‘ƒ) = (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(.rβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))))
12 eqid 2733 . . . 4 ( ·𝑠 β€˜π‘ƒ) = ( ·𝑠 β€˜π‘ƒ)
131, 4, 5, 6, 3, 2, 12prdsvsca 17347 . . 3 (πœ‘ β†’ ( ·𝑠 β€˜π‘ƒ) = (𝑓 ∈ (Baseβ€˜π‘†), 𝑔 ∈ 𝐡 ↦ (π‘₯ ∈ 𝐼 ↦ (𝑓( ·𝑠 β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))))
14 eqidd 2734 . . 3 (πœ‘ β†’ (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (𝑆 Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))) = (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (𝑆 Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))))
15 eqidd 2734 . . 3 (πœ‘ β†’ (∏tβ€˜(TopOpen ∘ 𝑅)) = (∏tβ€˜(TopOpen ∘ 𝑅)))
16 eqid 2733 . . . 4 (leβ€˜π‘ƒ) = (leβ€˜π‘ƒ)
171, 4, 5, 6, 3, 16prdsle 17349 . . 3 (πœ‘ β†’ (leβ€˜π‘ƒ) = {βŸ¨π‘“, π‘”βŸ© ∣ ({𝑓, 𝑔} βŠ† 𝐡 ∧ βˆ€π‘₯ ∈ 𝐼 (π‘“β€˜π‘₯)(leβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))})
18 eqid 2733 . . . 4 (distβ€˜π‘ƒ) = (distβ€˜π‘ƒ)
191, 4, 5, 6, 3, 18prdsds 17351 . . 3 (πœ‘ β†’ (distβ€˜π‘ƒ) = (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ sup((ran (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(distβ€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < )))
20 eqidd 2734 . . 3 (πœ‘ β†’ (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))) = (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯))))
21 eqidd 2734 . . 3 (πœ‘ β†’ (π‘Ž ∈ (𝐡 Γ— 𝐡), 𝑐 ∈ 𝐡 ↦ (𝑑 ∈ ((2nd β€˜π‘Ž)(𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))β€˜π‘Ž) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘…β€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯))))) = (π‘Ž ∈ (𝐡 Γ— 𝐡), 𝑐 ∈ 𝐡 ↦ (𝑑 ∈ ((2nd β€˜π‘Ž)(𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))β€˜π‘Ž) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘…β€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯))))))
221, 2, 3, 7, 9, 11, 13, 14, 15, 17, 19, 20, 21, 4, 5prdsval 17342 . 2 (πœ‘ β†’ 𝑃 = (({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (+gβ€˜π‘ƒ)⟩, ⟨(.rβ€˜ndx), (.rβ€˜π‘ƒ)⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘†βŸ©, ⟨( ·𝑠 β€˜ndx), ( ·𝑠 β€˜π‘ƒ)⟩, ⟨(Β·π‘–β€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (𝑆 Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))))⟩}) βˆͺ ({⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ 𝑅))⟩, ⟨(leβ€˜ndx), (leβ€˜π‘ƒ)⟩, ⟨(distβ€˜ndx), (distβ€˜π‘ƒ)⟩} βˆͺ {⟨(Hom β€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))⟩, ⟨(compβ€˜ndx), (π‘Ž ∈ (𝐡 Γ— 𝐡), 𝑐 ∈ 𝐡 ↦ (𝑑 ∈ ((2nd β€˜π‘Ž)(𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))β€˜π‘Ž) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘…β€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯)))))⟩})))
23 prdstset.l . 2 𝑂 = (TopSetβ€˜π‘ƒ)
24 tsetid 17239 . 2 TopSet = Slot (TopSetβ€˜ndx)
25 fvexd 6858 . 2 (πœ‘ β†’ (∏tβ€˜(TopOpen ∘ 𝑅)) ∈ V)
26 snsstp1 4777 . . . 4 {⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ 𝑅))⟩} βŠ† {⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ 𝑅))⟩, ⟨(leβ€˜ndx), (leβ€˜π‘ƒ)⟩, ⟨(distβ€˜ndx), (distβ€˜π‘ƒ)⟩}
27 ssun1 4133 . . . 4 {⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ 𝑅))⟩, ⟨(leβ€˜ndx), (leβ€˜π‘ƒ)⟩, ⟨(distβ€˜ndx), (distβ€˜π‘ƒ)⟩} βŠ† ({⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ 𝑅))⟩, ⟨(leβ€˜ndx), (leβ€˜π‘ƒ)⟩, ⟨(distβ€˜ndx), (distβ€˜π‘ƒ)⟩} βˆͺ {⟨(Hom β€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))⟩, ⟨(compβ€˜ndx), (π‘Ž ∈ (𝐡 Γ— 𝐡), 𝑐 ∈ 𝐡 ↦ (𝑑 ∈ ((2nd β€˜π‘Ž)(𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))β€˜π‘Ž) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘…β€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯)))))⟩})
2826, 27sstri 3954 . . 3 {⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ 𝑅))⟩} βŠ† ({⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ 𝑅))⟩, ⟨(leβ€˜ndx), (leβ€˜π‘ƒ)⟩, ⟨(distβ€˜ndx), (distβ€˜π‘ƒ)⟩} βˆͺ {⟨(Hom β€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))⟩, ⟨(compβ€˜ndx), (π‘Ž ∈ (𝐡 Γ— 𝐡), 𝑐 ∈ 𝐡 ↦ (𝑑 ∈ ((2nd β€˜π‘Ž)(𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))β€˜π‘Ž) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘…β€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯)))))⟩})
29 ssun2 4134 . . 3 ({⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ 𝑅))⟩, ⟨(leβ€˜ndx), (leβ€˜π‘ƒ)⟩, ⟨(distβ€˜ndx), (distβ€˜π‘ƒ)⟩} βˆͺ {⟨(Hom β€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))⟩, ⟨(compβ€˜ndx), (π‘Ž ∈ (𝐡 Γ— 𝐡), 𝑐 ∈ 𝐡 ↦ (𝑑 ∈ ((2nd β€˜π‘Ž)(𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))β€˜π‘Ž) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘…β€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯)))))⟩}) βŠ† (({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (+gβ€˜π‘ƒ)⟩, ⟨(.rβ€˜ndx), (.rβ€˜π‘ƒ)⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘†βŸ©, ⟨( ·𝑠 β€˜ndx), ( ·𝑠 β€˜π‘ƒ)⟩, ⟨(Β·π‘–β€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (𝑆 Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))))⟩}) βˆͺ ({⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ 𝑅))⟩, ⟨(leβ€˜ndx), (leβ€˜π‘ƒ)⟩, ⟨(distβ€˜ndx), (distβ€˜π‘ƒ)⟩} βˆͺ {⟨(Hom β€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))⟩, ⟨(compβ€˜ndx), (π‘Ž ∈ (𝐡 Γ— 𝐡), 𝑐 ∈ 𝐡 ↦ (𝑑 ∈ ((2nd β€˜π‘Ž)(𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))β€˜π‘Ž) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘…β€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯)))))⟩}))
3028, 29sstri 3954 . 2 {⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ 𝑅))⟩} βŠ† (({⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), (+gβ€˜π‘ƒ)⟩, ⟨(.rβ€˜ndx), (.rβ€˜π‘ƒ)⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘†βŸ©, ⟨( ·𝑠 β€˜ndx), ( ·𝑠 β€˜π‘ƒ)⟩, ⟨(Β·π‘–β€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (𝑆 Ξ£g (π‘₯ ∈ 𝐼 ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))))⟩}) βˆͺ ({⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ 𝑅))⟩, ⟨(leβ€˜ndx), (leβ€˜π‘ƒ)⟩, ⟨(distβ€˜ndx), (distβ€˜π‘ƒ)⟩} βˆͺ {⟨(Hom β€˜ndx), (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))⟩, ⟨(compβ€˜ndx), (π‘Ž ∈ (𝐡 Γ— 𝐡), 𝑐 ∈ 𝐡 ↦ (𝑑 ∈ ((2nd β€˜π‘Ž)(𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))𝑐), 𝑒 ∈ ((𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ Xπ‘₯ ∈ 𝐼 ((π‘“β€˜π‘₯)(Hom β€˜(π‘…β€˜π‘₯))(π‘”β€˜π‘₯)))β€˜π‘Ž) ↦ (π‘₯ ∈ 𝐼 ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘…β€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯)))))⟩}))
3122, 23, 24, 25, 30prdsbaslem 17340 1 (πœ‘ β†’ 𝑂 = (∏tβ€˜(TopOpen ∘ 𝑅)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  Vcvv 3444   βˆͺ cun 3909  {csn 4587  {cpr 4589  {ctp 4591  βŸ¨cop 4593   ↦ cmpt 5189   Γ— cxp 5632  dom cdm 5634   ∘ ccom 5638  β€˜cfv 6497  (class class class)co 7358   ∈ cmpo 7360  1st c1st 7920  2nd c2nd 7921  Xcixp 8838  ndxcnx 17070  Basecbs 17088  +gcplusg 17138  .rcmulr 17139  Scalarcsca 17141   ·𝑠 cvsca 17142  Β·π‘–cip 17143  TopSetcts 17144  lecple 17145  distcds 17147  Hom chom 17149  compcco 17150  TopOpenctopn 17308  βˆtcpt 17325   Ξ£g cgsu 17327  Xscprds 17332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-map 8770  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-sup 9383  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-3 12222  df-4 12223  df-5 12224  df-6 12225  df-7 12226  df-8 12227  df-9 12228  df-n0 12419  df-z 12505  df-dec 12624  df-uz 12769  df-fz 13431  df-struct 17024  df-slot 17059  df-ndx 17071  df-base 17089  df-plusg 17151  df-mulr 17152  df-sca 17154  df-vsca 17155  df-ip 17156  df-tset 17157  df-ple 17158  df-ds 17160  df-hom 17162  df-cco 17163  df-prds 17334
This theorem is referenced by:  prdshom  17354  prdsco  17355  prdstopn  22995  prdstps  22996
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